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July  2013, 12(4): 1745-1753. doi: 10.3934/cpaa.2013.12.1745

Remarks on nonlinear equations with measures

Moshe Marcus 1,

1. 

Department of Mathematics, Technion, Haifa 32000, Israel

Received  February 2011 Revised  July 2012 Published  November 2012

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Omega$.
Keywords: Orlicz - Sobolev spaces, absorption term, Semi-linear elliptic equation, weak solution, capacities..
Mathematics Subject Classification: Primary: 35J65, 46E30; Secondary: 46E35, 32U2.
Citation: Moshe Marcus. Remarks on nonlinear equations with measures. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1745-1753. doi: 10.3934/cpaa.2013.12.1745
References:
[1]

D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'', Grundlehren Math. Wissen., 314 (1996).   Google Scholar

[2]

N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz,, Ann. Sci. Math. Qu\'ebec, 12 (1994), 1.   Google Scholar

[3]

P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires,, Ann. Inst. Fourier, 34 (1984), 185.   Google Scholar

[4]

D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data,, Ann. I. H. Poincar\'e - AN, 22 (2005), 799.   Google Scholar

[5]

Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation,, J. Evolution Eq., 3 (2003), 673.   Google Scholar

[6]

H. Brezis, Notes, unpublished (circa 1970)., (1970).   Google Scholar

[7]

H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.   Google Scholar

[8]

Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems,, J. Funct. An., 8 (1971), 52.   Google Scholar

[9]

M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'', P. Noordhoff, (1961).   Google Scholar

[10]

J. B. Keller, On solutions of $\Delta u = f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.   Google Scholar

[11]

M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case,, Arch. Rat. Mech. Anal., 144 (1998), 201.   Google Scholar

[12]

M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption,, J. European Math. Soc., 6 (2004), 483.   Google Scholar

[13]

R. Osserman, On the inequality $\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641.   Google Scholar

[14]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'', Pitman Research Notes, 353 (1996).   Google Scholar

show all references

References:
[1]

D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'', Grundlehren Math. Wissen., 314 (1996).   Google Scholar

[2]

N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz,, Ann. Sci. Math. Qu\'ebec, 12 (1994), 1.   Google Scholar

[3]

P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires,, Ann. Inst. Fourier, 34 (1984), 185.   Google Scholar

[4]

D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data,, Ann. I. H. Poincar\'e - AN, 22 (2005), 799.   Google Scholar

[5]

Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation,, J. Evolution Eq., 3 (2003), 673.   Google Scholar

[6]

H. Brezis, Notes, unpublished (circa 1970)., (1970).   Google Scholar

[7]

H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.   Google Scholar

[8]

Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems,, J. Funct. An., 8 (1971), 52.   Google Scholar

[9]

M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'', P. Noordhoff, (1961).   Google Scholar

[10]

J. B. Keller, On solutions of $\Delta u = f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.   Google Scholar

[11]

M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case,, Arch. Rat. Mech. Anal., 144 (1998), 201.   Google Scholar

[12]

M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption,, J. European Math. Soc., 6 (2004), 483.   Google Scholar

[13]

R. Osserman, On the inequality $\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641.   Google Scholar

[14]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'', Pitman Research Notes, 353 (1996).   Google Scholar

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